Each \(\tilde{\bf {z}}\in \widetilde{\mathcal {C}}_{\infty }\)  corresponds to a pair \((X_{P(\tilde{\bf {z}})}, \alpha )\)  where \(\alpha \)  is a (level 2) theta structure on the Riemann surface \(X_{P(\tilde{\bf {z}})}\) , see section 4.5 of [1]} for details. A theta structure on \(X_{P(\tilde{\bf {z}})}\)  gives a canonical isomorphism \(\mathbb {P}(\mathcal {V}^{\scriptscriptstyle (}k\) )\(\) XP(z)) P(Sk(F2))\(, see (\ref {kiso}). This implies that each fiber of \) P*(P(V\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ))\( is canonically isomorphic to \) P(Sk(F2))\( and the proof follows.\)
Projectively flat connections
We recall the (Čech) description of a projectively flat connection, see section 2.3.3 and 2.3.6 of [1]}, page 369 of [3]}, and section 3.3 of [4]}.
Definition 2.5 
Let \(\mathbb {P}(V) \rightarrow M\)  be the projective bundle associated with a holomorphic vector bundle \(V \rightarrow M\)  over a complex manifold \(M\) . A projective connection in \(\mathbb {P}(V) \rightarrow M\)  is the data \(\lbrace U_i, \nabla _i\rbrace \) , where \(\lbrace U_i\rbrace \)  is an open covering of \(M\)  and \(\nabla _i\)  is a connection in the restriction \(V \vert _{U_i}\)  such that on any intersection \(U_i \cap U_j\)  the difference \(\nabla _i - \nabla _j\)  is scalar multiple of the identity, i.e.
\(\nabla _i - \nabla _j = \alpha \cdot \operatorname{Id}\) 
where \(\alpha \)  is a holomorphic 1-form on \(U_i \cap U_j\)  and \(\operatorname{Id}\)  is the multiplicative identity of the ring \(\Gamma (U_i \cap U_j, \operatorname{End}(V)_{|U_i \cap U_j})\) . Moreover, a projective connection \(\lbrace U_i, \nabla _i\rbrace \)  is a projectively flat connection if each \(\nabla _i\)  is flatUsually, a weaker condition that each \(\nabla _i\)  is projectively flat is imposed..

Two projective connections \(\lbrace U_i, \nabla _i\rbrace \)  and \(\lbrace U^{\prime }_i, \nabla ^{\prime }_i\rbrace \)  are called equivalent if on any intersection \(U_i \cap U^{\prime }_j\)  the difference \(\nabla _i - \nabla ^{\prime }_j\)  is a scaler multiple of the identity. If \(G\)  is a group acting by automorphisms on \(M\) , then a projective connection \(\lbrace U_i, \nabla _i\rbrace \)  is invariant under the action of \(G\)  if \(\lbrace U_i, \nabla _i\rbrace \)  and \(\lbrace g^*(U_i), g^*(\nabla _i)\rbrace \)  are equivalent for all \(g \in G\) .
A flat connection in a holomorphic vector bundle \(V \rightarrow M\)  induces a unique projectively flat connection in \(\mathbb {P}(V) \rightarrow M\) . Two flat connections induce equivalent projectively flat connections if their difference is a scaler multiple of the identity.
Upon the choice of a base point \(x \in M\) , the monodromy of a projectively flat connection provides a group homomorphism
\(\mathfrak {M} \colon \pi _1(M, x) \rightarrow \operatorname{Aut}(\mathbb {P}V_x),\) 
see the before mentioned references for details.
Recall the open subset \(\mathcal {C}:= \lbrace (z_1, \dots , z_6) \in \mathcal {C}_{\infty } \mid z_i \ne \infty \rbrace \) . We outline the construction of the flat connection, see (REF ), in the bundle
\(S^k(F_2) \times \mathcal {C}\) 
carried out in [1]}. This connection is then used to prove proposition REF  and corollary REF .
Denote the delta function basis of \(F_2\)  by \(\langle x_1, x_2, x_3, x_4 \rangle \) . Let \(\partial _i \)  be the differential operator defined by
\(\partial _i (x_j)= \delta _{ij}:=\left\lbrace \begin{array}{l l}1 \,\,\,\text{if $i=j$}\\0 \,\,\, \text{if $i \ne j$ }\end{array} \right.\) 
For all \(1 \leqslant i < j \leqslant 6 \)  consider the differential operators
\(&\Omega ^{1,2}:= -(x_1\partial _1 + x_2\partial _2 - x_3\partial _3 - x_4\partial _4)^2 \quad \quad \Omega ^{1,3}:= -(x_1\partial _4 - x_2\partial _3 - x_3\partial _2 + x_4\partial _1)^2\\&\Omega ^{1,4} := +(x_1\partial _4 + x_2\partial _3 - x_3\partial _2 - x_4\partial _1)^2\quad \quad \Omega ^{1,5} := +(x_1\partial _3 - x_2\partial _4 - x_3\partial _1 + x_4\partial _2) ^2\\&\Omega ^{1,6} := -(x_1\partial _3 + x_2\partial _4 + x_3\partial _1 + x_4\partial _2)^2\quad \quad \Omega ^{2, 3} := +(x_1\partial _4 - x_2\partial _3 + x_3\partial _2 - x_4\partial _1)^2\\&\Omega ^{2, 4} := -(x_1\partial _4 + x_2\partial _3 + x_3\partial _2 + x_4\partial _1)^2\quad \quad \Omega ^{2, 5} := -(x_1\partial _3 - x_2\partial _4 + x_3\partial _1 - x_4\partial _2)^2\\&\Omega ^{2, 6} := +(x_1\partial _3 + x_2\partial _4 - x_3\partial _1 - x_4\partial _2)^2\quad \quad \Omega ^{3, 4} := -(x_1\partial _1 - x_2\partial _2 + x_3\partial _3 - x_4\partial _4)^2\\&\Omega ^{3, 5} := -(x_1\partial _2 + x_2\partial _1 + x_3\partial _4 + x_4\partial _3)^2\quad \quad \Omega ^{3, 6} := +(x_1\partial _2 - x_2\partial _1 - x_3\partial _4 + x_4\partial _3)^2\\&\Omega ^{4, 5} := +(x_1\partial _2 - x_2\partial _1 + x_3\partial _4 - x_4\partial _3)^2\quad \quad \Omega ^{4, 6} := -(x_1\partial _2 + x_2\partial _1 - x_3\partial _4 - x_4\partial _3)^2\\&\Omega ^{5, 6} := -(x_1\partial _1 - x_2\partial _2 - x_3\partial _3 + x_4\partial _4)^2,\) 

where the composition law is
\((x_i\partial _j) \circ (x_k \partial _l)= x_ix_k\partial _j \partial _l +\delta _{jk}x_i\partial _l.\) 
Let \(\widehat{\Omega }^{i,j}\)  denote the degree two part of \(\Omega ^{i,j}\) .
The operator (REF ) has an induced action on any monomial \(x_1^ax_2^bx_3^cx_4^d\) , see section 4.5 of [1]}. For a positive integer \(k\) , choose the basis of \(S^k(F_2)\)  given by monomials \(x_1^ax_2^bx_3^cx_4^d\)  such that \(a+b+c+d=k\) . For all \(1 \leqslant i < j \leqslant 6\) , the induced action of the operator \(\widehat{\Omega }^{i,j}\)  on \(S^k(F_2)\)  preserves \(S^k(F_2)\) , see section 3.2.5 of [1]}, and we denote this induced action by \(\widehat{\Omega }^{i,j}_k \in \operatorname{End}(S^k(F_2))\) . These endomorphisms satisfy the infinitesimal braid relations, i.e.
\([\widehat{\Omega }^ {s,t}_k, \widehat{\Omega }^ {u,v}_k]=0, \quad \quad [\widehat{\Omega }^ {s,u}_k, \widehat{\Omega }^ {s,t}_k + \widehat{\Omega }^ {t,u}_k] = 0\) 
where \(s, t, u, v\)  are all distinct, see proposition 3.2.4 of [1]}.
For a positive integer \(k\) , let
\(\hspace{-5.0pt}k:= {-1\over 16(k+2)},\) 
and consider the following \(\operatorname{End}(S^k(F_2))\)  valued holomorphic 1-form on \(\mathcal {C}\) 

\(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
:=k1 i<j 6 i,jk dzi - dzj zi - zj.

Since (REF ) is a trivial vector bundle, the expression

\(\nabla _0^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 := d + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ,

where \(d\)  denotes the trivial connection, defines a connection in (REF ). Moreover, using the relations (REF ) and Arnold's trick, see proposition XIX.2.1 of [9]}, it can be shown that the curvature 2-form of the connection (REF ) vanishes, i.e.

\(-d\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  = 0.

We note the following relations, called the Ward identities in two-dimensional conformal field theory, see [10]}, which will be used later.
Proposition 2.6 
For a positive integer \(k\) , we have that

\(&\qquad \quad 1) \,\sum _{i < j} \widehat{\Omega }^{i,j}_k = 3\hspace{-5.0pt}kk^2 \operatorname{Id}^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
         2)  i j i,jk = kk2Id\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)        for any fixed \(j\)
         3)  i < j i,jk (zi + zj)= 6kk2Id\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  i < j zi + zj

where \(\operatorname{Id}^{\scriptscriptstyle (}k\) )\(\)\( is the multiplicative identity of the ring \) End(Sk(F2))\(.\)
Direct computations, notice that \((2)\)  implies \((3)\) .
The automorphism group of \(^1\)  acts on \(\mathcal {C}_{\infty }\)  by automorphisms. It is well-known that the group of automorphisms of \(^1\)  is isomorphic to \(\mathbb {P}\mathrm {SL}(2, \)  and generated
by the following three transformations, see page 10 of [11]},
\(T(z)= z + a, \quad D(z) = bz, \quad and \quad I(z)={1\over z}\) 
where \(z \in ^1\) , \(a \in , and \) b *\(. These three transformations are called translations, dilations, and inversion respectively.\)
Proposition 2.7 
The connection (REF ) in the bundle (REF ) determines a unique (up to equivalence) projectively flat connection \(\nabla ^{\scriptscriptstyle (}k\) )\(\) := {Ui, \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) i}\( in the projective bundle\begin{equation}\mathbb {P}( S^k(F_2)) \times \mathcal {C}_{\infty }.\end{equation}Moreover, this projectively flat connection \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\( is invariant under the \) PSL(2, \( action on \) C\(.\)
We must produce the data \(\lbrace U_i, \nabla ^{\scriptscriptstyle (}k\) )\(\) i}\( satisfying the conditions of definition \ref {pfc}. It is easy to construct an open covering \) {Ui}\( of \) C\( such that each \) Ui = i*(C)\( (WE NEED TO GIVE \) i's\() for some \) i PSL(2, \(. Let \) 0\( be the identity. Then \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0\(, defined in (\ref {hitchinc}), is a flat connection in \) Sk(F2) U0\(. For each \) Ui\(, we define\begin{equation}\nabla ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\) i := i*(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0) = d + i*(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )

which gives a connection in the restriction \(S^k(F_2) \times U_i\) . Moreover, since \(\tau _i\)  is a biholomorphism, the curvature of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) i\( is the pull back of the curvature of \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0\(, since the curvature of \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0\( vanishes, it follows that \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) i\( is flat. To show that \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) i - \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) j\( is a scaler multiple of the identity on the intersections \) Ui Uj\(, we compute that on the intersection \) C*(C)\(,\)\(\nabla ^{\scriptscriptstyle (}k\) )\(\) 0 - *(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0) = 0
\(when \) 
\( is a translation or a dilation, and{\begin{@align*}{1}{-1}\nabla ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{@align*}}_0 - \tau ^*(\nabla ^{\scriptscriptstyle (}k\) )\(\) 0) = k1 i < j 6 i,jk(dzi zi + dzj zj)
= 6kkId\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  1 i < j 6 (dzi zi + dzj zj)

when \(\tau \)  is the inversion. The second equality follows from identity (3) of proposition REF , notice also that the 1-form \(6\hspace{-5.0pt}kk\sum _{1 \leqslant i < j \leqslant 6} \big ({dz_i \over z_i} + {dz_j \over z_j}\big )\)  is holomorphic on the intersection. These three cases suffice since \(\mathbb {P}\mathrm {SL}(2, \)  is generated by translations, dilations, and the inversion.
Moreover, these calculations also show that two different choices of the set of automorphisms \(\lbrace \tau _i\rbrace \)  give equivalent projectively flat connections, thus the connection (REF ) uniquely determines (up to equivalence) the projectively flat connection. Lastly, by its construction, it is clear that \(\lbrace U_i, \nabla ^{\scriptscriptstyle (}k\) )\(\) i}\( is invariant under the \) PSL(2, \( action.\) 
Recall the covering \({P} \colon \widetilde{\mathcal {C}}_{\infty } \rightarrow \mathcal {C}_{\infty }\) .
Corollary 2.8 
The pull back 
\({P}^*(\nabla ^{\scriptscriptstyle (}k\) 
)\(\) )={P*(Ui), P*(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) i)}
\( defines a unique projectively flat connection in \) 
P*(P(V\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )\(.\)
Follows from lemma REF .

Teichmüller curves and pseudo-Anosov mapping classes
In this section we recall some basic results in Teichmüller theory and Teichmüller curves. We end this section with a discussion of our main object of interest: the Veech curve. Let \(S_g\)  be a closed, connected and oriented surface of genus \(g \geqslant 2\) . A marked Riemann surface is a pair \((X, f)\)  where \(X\)  is a smooth compact connected Riemann surface and \(f \colon S_g \rightarrow X\)  is an orientation preserving diffeomorphism. Let
\(\mathcal {T}_g := (X, f) / \sim ,\) 
where \((X, f) \sim (X^{\prime }, f^{\prime })\)  if there exists a biholomorphism \(g \colon X \rightarrow X^{\prime }\)  such that \( g \circ f\)  is isotopic to \(f^{\prime }\) . The group \(\Gamma _g\)  acts on \(\mathcal {T}_g\)  as follows. For any \([\Psi ] \in \Gamma _g\)  let
\([\Psi ] \cdot [(X, f)] := [(X, f \circ \Psi ^{-1})].\) 
The class \([(X, f \circ \Psi ^{-1})]\)  is independent of the choice of the representative in \([\Psi ]\)  as any two isotopic markings are equivalent, this implies that (REF ) gives a well defined group action of \(\Gamma _g\)  on \(\mathcal {T}_g\) .
\(\mathcal {T}_g\)  has the structure of a complex manifold which is homeomorphic to an open ball in \({3g-3}\) . Traditional approach to constructing the complex structure is by Bers embedding of \(\mathcal {T}_g\)  into the complex vector space of holomorphic quadratic differentials on a (reference) compact Riemann surface of genus \(g\) , see chapter 6 of [12]} for details. A differential geometric approach to the complex structure is given by the Weil-Petersson metric, see section 3 and theorem 3.3 in [13]}. In the algebro-geometric approach one shows that for all \([(X, f)] \in \mathcal {T}_g\)  there exists a neighbourhood which can be identified with the base space of a Kuranishi family of (equivalence classes of) marked compact Riemann surfaces of genus \(g\)  with central fiber isomorphic to \([(X, f)]\) . By the universal property of these Kuranishi families, and the fact that a marked Riemann surface has trivial automorphisms, these local neighborhoods can be glued together into a single Kuranishi family, thus giving \(\mathcal {T}_g\)  the structure of a complex manifold, see [14]} chapter XV for details. In all the three topologies induced by these three complex structures, it can be shown that \(\mathcal {T}_g\)  is an open ball in \({3g-3}\) . From now on we consider \(\mathcal {T}_g\)  as a complex manifold with complex structure induced from the Bers embedding.
H. L. Royden, see [15]}, showed that the action (REF ) of \(\Gamma _g\)  on \(\mathcal {T}_g\)  is by biholomorphic automorphisms and the group of biholomorphic automorphims of \(\mathcal {T}_g\) , for genus \(g\geqslant 3\) , can be identified with \(\Gamma _g.\)  The group of biholomorphic automorphims of \(\mathcal {T}_2\)  is equivalent to \(\Gamma _2\)  modulo the hyperelliptic involution. Moreover, the action of \(\Gamma _g\)  on \(\mathcal {T}_g\)  is properly discontinuous but not free, see theorem 6.18 in [12]}, or proposition 1.13 in [13]} for the proof. The quotient
\(\mathcal {M}_g := \mathcal {T}_g / \Gamma _g\) 
is the moduli space of compact Riemann surfaces of genus \(g\) . Since the action (REF ) is not free, the space \(\mathcal {M}_g\)  is a complex orbifold of dimension \(3g-3\) , the orbifold singularities correspond to (equivalence classes of) Riemann surfaces which admit non-trivial automorphisms. Since \(\mathcal {T}_g\)  is simply connected, we have the identification
\(\Gamma _g \cong \pi _1^{orb}(\mathcal {M}_g, *).\) 
Like any complex manifold, \(\mathcal {T}_g\)  can be equipped with the Kobayashi pseudo-metric, see [18]} for a comprehensive introduction, see also page 167 of [12]}. If the Kobayashi pseudo-metric is complete (or non-degenerate), see page 168 of [18]}, then the complex manifold is called Kobayashi hyperbolic, or a hyperbolic complex space.
More classically, a metric on \(\mathcal {T}_g\)  is introduced by using (the existence and uniqueness of) minimal dilation quasi-conformal maps between compact Riemann surfaces equipped with holomorphic quadratic differentials of the same typei.e, the number and the multiplicity of the zeroes of the quadratic differentials match, this metric is called the Teichmüller metric, see section 5.1.3 of [12]} for more details. H. L Royden showed that the Kobayashi pseudo-metric is equivalent to the Teichmüller metric, see [15]} or theorem 6.21 in [12]}, and since the Teichmüller metric is complete, see theorem 5.4 in [12]}, it follows that \(\mathcal {T}_g\)  is Kobayashi hyperbolic.
Since the Kobayashi metric is always invariant under the group of biholomorphic automorphisms, see [18]} or [12]}, the complex orbifold \(\mathcal {M}_g\)  is also Kobayashi hyperbolic. Let \(V\)  be a hyperbolic Riemann surface, i.e. \(V \cong \mathbb {H}/G\)  where \(G\)  is a discrete subgroup of \(\mathbb {P}\mathrm {SL}(2, \mathbb {R})\) , and let
\(\phi \colon V \rightarrow \mathcal {M}_g\) 
be a generically injective holomorphic map. The pair \((V, \phi )\)  is called a Teichmüller curve if \(V\)  has finite hyperbolic area and \(\phi \)  is an isometry with respect to the Kobayashi metric on the domainOn \(\mathbb {H}\) , and thus on \(V\) , the Kobayashi metric is equivalent to the Poincaré metric normalized such that its curvature is -4, see page 20 of [18]} and section 1 of [28]}. and the codomain. In other words, a Teichmüller curve is an irreducible algebraic curve in \(\mathcal {M}_g\)  which is a totally geodesic sub manifold for the Kobayashi (=Teichmüller) metric.
The simplest example of a Teichmüller curve is \((\mathcal {M}_1, \operatorname{Id})\) , where \(\mathcal {M}_1\)  is the moduli space of compact Riemann surfaces of genus one. For \(g \geqslant 2\) , the existence of Teichmüller curves already follows from W. Thurston's construction of pseudo-Anosov mapping classes, see section 8 of [29]}. From [30]} it follows that for any \(g \geqslant 2\)  there exist countably many Teichmüller curves in \(\mathcal {M}_g\) . C. T. McMullen, see [31]}, constructed an infinite and explicit family of (primitive)i.e. These Teichmüller curves don't cover Teichmüller curves in lower genera. Teichmüller curves in \(\mathcal {M}_2\) , he also discovered that the image of any Teichmüller curve in \(\mathcal {M}_2\)  always lies on a Hilbert modular surfaceHilbert modular surfaces, see Hirzebruch's seminal paper [32]} and the book [33]} for a general introduction, live in \(\mathcal {A}_2\) , the moduli space of abelian surfaces. Here we mean the image of the Teichmüller curve under the Torelli map \(\mathcal {M}_2 \rightarrow \mathcal {A}_2\)  always lies in a Hilbert modular surface.. He then extended his construction to genus \(g=3 \,\text{and} \,4\)  in [34]}.
A complete classification of (primitive) Teichmüller curves in \(\mathcal {M}_2\)  follows from the work of [31]}, [36]}, [28]} and [38]}, we refer to section 5.5 in [39]} for more details. W. Veech, in [40]}, constructed two Teichmüller curves for each \(g \geqslant 2\) , the image of both of these lie in the hyperelliptic locus of \(\mathcal {M}_g\) . In [41]}, using algebro-geometric methods a large number of Teichmüller curves are produced, the strong point of this approach being that the family of Riemann surfaces that these Teichmüller curves parametrize can be described explicitly as a (complex) one parameter family of polynomials. For further details on this approach we refer to the excellent article [39]}. More recently, the authors in [43]} have innovated a new description of (certain) Teichmüller curves in terms of theta functions, from their viewpoint these Teichmüller curves lead to a genuinely new type of modular forms.
Typical construction of Teichmüller curves comes from Veech surfaces. If \((X, f)\)  is a marked Riemann surface, then a holomorphic quadratic differential \(q \in H^0(X, (\Omega ^1_X)^{\otimes 2})\)  generates an embedding
\(\widetilde{\phi }_{(X, q)} \colon \mathbb {H}\rightarrow \mathcal {T}_g,\) 
which is an isometry for the Kobayashi metrics on the domain and the codomain. The image \(\widetilde{\phi }_{(X, q)}(\mathbb {H})\)  is a complex geodesic for the Kobayashi metric on \(\mathcal {T}_g\) . Such an embedding is called the Teichmüller disc generated by the pair \((X, q)\)  Since the marking does not play any role in what follows, we drop it from the notation. A pair \((X, q)\)  is also called a half translation surface., see [29]} and [40]} for the construction of such discs. Let \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  be the subgroup of \(\Gamma _g\)  which preserves the image \(\widetilde{\phi }_{(X, q)}(\mathbb {H})\)  set wise. The embedding (REF ), upon passing to the quotient by the action of \(\Gamma _g\) , induces the complex geodesic
\(f \colon \mathbb {H}\rightarrow \mathcal {T}_g \rightarrow \mathcal {M}_g\) 
which factors through \(\mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\) . If \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  is a lattice in \(\operatorname{Aut}(\widetilde{\phi }_{(X, q)}(\mathbb {H}))(=\mathbb {P}\mathrm {SL}(2,\mathbb {R}))\) , i.e. \(\mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  has finite hyperbolic area, then the induced map, which we also denote by \(f\) ;
\(f \colon \mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)}) \rightarrow \mathcal {M}_g\) 
is proper and generically injective. It follows that the pair \((\mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)}), f)\)  is a Teichmüller curve. For further details on the structure of the group \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\) , several examples of groups which arise as \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\) , and open problems related to the group \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  we refer to the excellent survey article [46]}. A pair \((X, q)\)  such that \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  is a lattice is called a Veech surface. Veech surfaces are intimately related with polygonal billiard tables which have optimal dynamics, for dynamical aspects of Veech surfaces we refer to [40]}.
Assume that \((\mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)}), f)\)  is a Teichmüller curve. Our interest in \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  lies in the fact that this subgroup of \(\Gamma _g\)  contains infinitely many pseudo-Anosov mapping classes which can be described explicitly. Denote by \(V:= \mathbb {H}/\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\) , and let \(\gamma \)  be a closed geodesic (with respect to the hyperbolic metric) in \(V\) . Since \(\pi _1^{orb}(V) \cong \operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  it follows that the homotopy class \([\gamma ] \in \pi _1^{orb}(V)\)  corresponds to a unique (up to conjugation) element in \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\) , denote this element by \(\varphi _{\gamma }\) .
Let \(\widetilde{\gamma }\)  be a lift of \(\gamma \)  such that \(\widetilde{\gamma } \subset \widetilde{\phi }_{(X, q)}(\mathbb {H})\) , c.f. (REF ). From the theory of hyperbolic Riemann surfaces, we know that the action of \(\varphi _{\gamma }\)  on \(\widetilde{\phi }_{(X, q)}(\mathbb {H})\)  preserves the geodesic \(\widetilde{\gamma }\) .
L. Bers, see section 6.5 of [12]}, showed that an element of \(\Gamma _g\)  is pseudo-Anosov if and only if its action on \(\mathcal {T}_g\)  preserves a (real) geodesic with respect to the Teichmüller, or equivalently Kobayashi, metric. By construction, \(\widetilde{\gamma }\)  is a (real) geodesic with respect to the Kobayashi metric which is preserved by \(\varphi _{\gamma }\) . It follows that \(\varphi _{\gamma }\)  is a pseudo-Anosov element in \(\Gamma _g\) . In this way, to any closed geodesic in \(V\)  we can associate a pseudo-Anosov element in \(\Gamma _g\) . Moreover, if the generators of \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  are known, then any pseudo-Anosov associated to a closed geodesic in \(V\)  can be written as a unique word (up to conjugation) in these generators.
Let \(n=2g+1\)  and \(n^{\prime }=2g+2\)  with \(g \geqslant 2\) . For all \(n\) , let \(\operatorname{Stab}({\widetilde{\phi }}_n):= \operatorname{Stab}({\widetilde{\phi }}_{(X_n, q_n)})\)  where \(X_n\)  is the unique compact hyperelliptic Riemann surface defined by the polynomial \(y^2 = x^n +1\)  and \(q_n = c_n {dx^2 \over y^2}\) , here \(c_n\)  is a positive constant such that \(\Vert q_n\Vert  =1\)see, for example, [40]} for the definition of this norm. Define \(\operatorname{Stab}({\widetilde{\phi }}_{n^{\prime }})\)  by replacing \(n\)  with \(n^{\prime }\)  in the definition of \(\operatorname{Stab}({\widetilde{\phi }}_n)\) . In [40]} it is shown that the group \(\operatorname{Stab}({\widetilde{\phi }_n})\)  is isomorphic to the Hecke triangle group of signature \((2, n, \infty )\) , and that the group \(\operatorname{Stab}({\widetilde{\phi }}_{n^{\prime }})\)  is isomorphic to the Hecke triangle group of signature \(({n^{\prime }\over 2}, \infty , \infty )\) .
Denote the Teichmüller curve corresponding to the lattice \(\operatorname{Stab}({\widetilde{\phi }_n})\)  by \((\mathchoice{\chi }{}{}{}\)\(\chi \) n, n)\(. An algebraic description of the map\begin{equation}\phi _n \colon \mathchoice{\chi }{}{}{}\end{equation}{\chi }\) n Mg,

given in [41]}, [28]}, and [53]}, is as follows. Let \(\widetilde{\mathchoice{\chi }{}{}{}}{\chi }\) n := (1-{n})\(, where \) n\( denotes the set of \) n\(^{\prime }th roots of unity. Let \) t\( be the restriction of the global rational coordinate on \) 1\( to \)\(\chi \)\(\chi \) n\( and let \) Dn Aut(\(\chi \)\(\chi \) n)\( denote the dihedral group generated by \) t n2t\( and \) t 1/ t\( where \) n := (2-1n)\(. For all \) t \(\chi \)\(\chi \) n\( let \) Ft\( be the unique compact hyperelliptic Riemann surface associated with the polynomial{\begin{@align*}{1}{-1}y^2= \prod _{i=0}^n (x - \zeta _n^{i} - t\zeta _n^{-i}).\end{@align*}}It turns out that \)\(\chi \)\(\chi \) n \(\chi \)\(\chi \) n/Dn\( and the isomorphism class of \) Yt\( depends only on \) [t] \(\chi \)\(\chi \) /D\(. The map (\ref {phin}) is then simply given, after realizing \)\(\chi \)\(\chi \) n\( as the quotient \)\(\chi \)\(\chi \) n/Dn\(, by \) n([t])= [Ft] Mg\(.\)
We restrict to the case \(g=2\)  and \(n=5\)  and drop \(n\)  as a subscript from the notation. A map \(\widetilde{\mathchoice{\chi }{}{}{}}{\chi }\)  C\( is given by\begin{equation}\tilde{\phi }(t) = (1+t, \zeta +\zeta ^{-1}t, \zeta ^2 +\zeta ^{-2}t, \zeta ^3 +\zeta ^{-3}t, \zeta ^4 +\zeta ^{-4}t, \infty ).\end{equation}\)
Definition 3.1 
Let \(S_n\)  be the symmetric group of degree \(n\) . Define \(\nu , \nu ^{\prime } \in S_5 \triangleleft S_6\)  by \(\nu (z_1,\dots , z_6) = (z_5, z_1, z_2, z_3, z_4, z_6)\)  and \(\nu ^{\prime }(z_1, \dots ,z_6)=(z_1, z_5, z_4, z_3, z_2, z_6).\)

It is clear from definition that \(\nu ^5=\nu ^{\prime 2}=\operatorname{Id}\) . We have the following relationship between \(\nu \)  and \(R\)  and between \(\nu ^{\prime }\)  and \(I\)  see [53]}.
Lemma 3.2 
\(\tilde{\phi }(R(t)) =\zeta \nu (\tilde{\phi }(t)) \)  and \(\tilde{\phi }(I(t))={1\over t}\nu ^{\prime }(\tilde{\phi }(t))\) . Notice that the second equality is only defined for \(t\)  not equal to 0 or \(\infty \) .

We compute
\(\zeta ^{-1}\tilde{\phi }(R(t))&= \zeta ^{-1}(1+\zeta ^2t, \zeta +\zeta t, \zeta ^2 + t, \zeta ^3 +\zeta ^{-1}t, \zeta ^4 +\zeta ^{-2}t, \infty )\\&= (\zeta ^{-1}+\zeta t, 1 + t, \zeta +\zeta ^{-1}t, \zeta ^2 +\zeta ^{-2}t, \zeta ^3 +\zeta ^{-3}t, \infty ).\) 

Now, the last tuple is the same as
\((\zeta ^4+\zeta ^{-4} t, 1 + t, \zeta +\zeta ^{-1}t, \zeta ^2 +\zeta ^{-2}t, \zeta ^3 +\zeta ^{-3}t, \infty ) = \nu (\tilde{\phi }(t)).\) 
By a similar calculation, we obtain that
\(t\tilde{\phi }(I(t)) = (1+t, \zeta ^{4} +\zeta ^{-4}t, \zeta ^3 +\zeta ^{-3}t, \zeta ^2 +\zeta ^{-2}t, \zeta +\zeta ^{-1}t, \infty )= \nu ^{\prime }(\tilde{\phi }(t)).\) 
Recall that \(\mathcal {M}_2\)  is the quotient of \(\mathcal {C}_{\infty }\)  by the product group \(\mathbb {P}\mathrm {SL}(2,  \times S_6\) . Since multiplication by \(\zeta \)  and \({1 \over t}\)  are both elements of the dilation subgroup of \(\mathbb {P}\mathrm {SL}(2, \)  this lemma immediately shows that the map () descends to give a well defined map \( \phi \colon \mathchoice{\chi }{}{}{}\)\(\chi \) M2\( and that we get the following commutative diagram\)
\(\begin{tikzpicture}[every node/.style={midway}][column sep={7em,between origins},row sep={3em}] at (0,0){ \node (R) {\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }}\end{tikzpicture} ; & \node (S) {\mathcal {C}_{\infty }}; \\\node (A) {\mathchoice{\chi }{}{}{}}{\chi }\) 
\(\)  ;  B) \(\mathcal {M}_2\) ;
;
[<-] (A) – (R) node[anchor=east] \(\pi _{\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }\)\(\) ;
[<-] (B) – (A) node[anchor=north] \({\phi }\) ;
[->] (R) – (S) node[anchor=south] \(\tilde{\phi }\) ;
[<-] (B) – (S)node[anchor=west] \(\pi _{\mathcal {M}_2}\) ;
Hitchin connection and Hyperlogarithms on the Veech curve 
Consider the space

\(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 
0 :=  10

and the embedding

\(\tilde{\psi }\colon \tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 
0 C

defined by

\(\tilde{\psi }(t) = \bigg ({1\over \zeta +\zeta ^{-1}t}, { 1 \over \zeta ^2 +\zeta ^{-2}t}, {1 \over \zeta ^3 +\zeta ^{-3}t}, {1 \over \zeta ^4 +\zeta ^{-4}t}, {1 \over 1+t}, 0 \bigg ), \quad \text{for}\, \text{all}\, t \in \tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 
0.
Lemma 3.3 
We have that \( \pi _{\mathcal {M}_2}(\tilde{\psi }(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 0)) (\(\chi \)\(\chi \) )\(.\)
Notice that \(\tilde{\psi }\)  is given by applying the involution \({1\over z} \in \mathbb {P}\mathrm {SL}(2, \)  and a permutation (which permutes the first and the second last coordinate) to \(\tilde{\phi }\) . Since \(\mathcal {M}_2\)  is defined as the quotient of \(\mathcal {C}_{\infty }\)  by the product group \(\mathbb {P}\mathrm {SL}(2,  \times S_6\) , it follows that \(\pi _{\mathcal {M}_2}(\tilde{\phi }(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) )) M2((\(\chi \)\(\chi \) 0))\( and by the commutativity of (\ref {CD}) we know that \) M2((\(\chi \)\(\chi \) ))= (\(\chi \)\(\chi \) )\(.\)
Proposition 3.4 
We compute that the pull back \(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\) )\(\) )\( is projectively equivalent to the \) End(Sk(F2))\( valued 1-form\begin{equation}\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\)\(\chi \)\(\chi \)  := k 1 i 5 Ai  dt t - i

where \(A_i = \widehat{\Omega }^{a,b} + \widehat{\Omega }^{c,d}\)  for \(1 \leqslant a<b,c<d \leqslant 5\)  such that \([a+b]=[c+d] = [i]\)  and \([x] := x \pmod {5}\) .

Remark 3.5 For any \(1 \leqslant i \leqslant 5\) , there exists only one solution for \(A_i\)  with the given constraints. From the expression it is obvious that the 1-form () has no poles at \(-\mu _5\)  and extends as a holomorphic 1-form to \(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\(, which justifies the subscript in (\ref {psipullback}).\)
Let \(D^0_6 \subset \mathcal {C}_{\infty }\)  be
\(D^0_6:=\lbrace  (z_1, \dots , z_6) \in \mathcal {C}| z_6 = 0\rbrace .\) 
Then \(\tilde{\psi }(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 0) D06\( and{\begin{@align*}{1}{-1}\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{@align*}}_{|D_6^0} &= {\hspace{-5.0pt}k}\bigg ( \sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} {dz_i - dz_j \over z_i - z_j} + \sum _{i=1}^5 \widehat{\Omega }^{i,6}{dz_i \over z_i}\bigg )\\&= {\hspace{-5.0pt}k}\bigg ( \sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} d \log (z_i - z_j) + \sum _{i=1}^5 \widehat{\Omega }^{i,6}d \log (z_i)\bigg ).\) 
Thus

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) = k( 1 i < j 5 i,j d (1 i + -it - 1 j + -jt)
- i=15 i,6d (i + -it)).

Now
\(d \log \bigg ({1 \over \zeta ^i + \zeta ^{-i}t} - {1 \over \zeta ^j + \zeta ^{-j}t}\bigg ) = &d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t) \\&- d\log (\zeta ^i + \zeta ^{-i}t) - d\log (\zeta ^j + \zeta ^{-j})\) 

which implies that

\(&\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) =k ( 1 i < j 5 i,jd (i + -it - j - -jt)
- 1 i < j 5 i,jd(i + -it)
- 1 i < j 5 i,j d(j + -jt)) -  i=15 i,6 d (i + -it)).

The last three terms can be simplified by
\(\sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j}d\log (\zeta ^i + \zeta ^{-i}t)+\sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} d\log (\zeta ^j + \zeta ^{-j}t)) + \sum _{ i=1}^5 \widehat{\Omega }^{i,6} d \log (\zeta ^i + \zeta ^{-i}t)=\) 
\(\sum _{1 \leqslant i < j \leqslant 6} \widehat{\Omega }^{i,j}d\log (\zeta ^i + \zeta ^{-i}t)+\sum _{1 \leqslant i < j \leqslant 6} \widehat{\Omega }^{i,j} d\log (\zeta ^j - \zeta ^{-j}t)) = -k^2 \operatorname{Id}\sum _{i=5}^6 d \log (\zeta ^i + \zeta ^{-i}t)\) 
where the last equality follows from (3) of propsition REF . This gives us that

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) =k ( 1 i < j 5 i,jd (i + -it - j - -jt) + k2 Idi=56 d (i + -it))

Now,
\(d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t)&={(\zeta ^{-i} - \zeta ^{-j}) dt \over (\zeta ^{i} - \zeta ^{j}) + (\zeta ^{-i} - \zeta ^{-j})t}\\&={dt \over {\zeta ^{i} - \zeta ^{j} \over \zeta ^{-i} - \zeta ^{-j}}+t} .\) 

This can be simplified by
\(-{\zeta ^{i} - \zeta ^{j} \over \zeta ^{-i} - \zeta ^{-j}} = {\zeta ^{i} - \zeta ^{j} \over \zeta ^{-j} - \zeta ^{-i}} {.} {\zeta ^ {i + j} \over \zeta ^ {i + j}} = \zeta ^{i +j},\) 
the result being
\(d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t)= {dt \over t - \zeta ^{i+j}}.\) 
Notice that
\(\zeta ^{i+j} = \zeta ^{k+l}\) 
if \(i+j=k+l (mod 5)\) .
Similarly,
\(d \log (\zeta ^i + \zeta ^{-i}t) &={ \zeta ^{-i} dt \over \zeta ^{i} + \zeta ^{-i}t}\\&= {dt \over t + \zeta ^{2i}}.\) 

This implies that

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) = k (1 i 5 Ai  dt t - i + k2 Id1i 5 dt t + 2i  )

which is projectively equivalent to ().
We define a connection in the trivial bundle \(H^0(^3, \mathcal {O}_{^3}(k)) \times \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( as follows\begin{equation}\nabla ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\)\(\chi \)\(\chi \)  := d + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\(\chi \)\(\chi \) .

The connection () can be considered as a meromorphic connection on \( with logarithmic singularities at \) 5\(. The residue at each singularity \) i 5\( is \) Ai\(. See \cite {De} for an excellent introduction to meromorphic connections with logarithmic singularities.\)
We give an explicit expression of the parallel transport of () along paths in \(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( in terms of iterated integrals of 1-forms on \)\(\chi \)\(\chi \)\(. The 1-forms \) dt t - i\(, where \) i 5\(, give a basis for \) H1dR(\(\chi \)\(\chi \) , \(. Let \) [a,b] \(\chi \)\(\chi \)\( be a smooth map and for each \) i\( let\begin{equation}f_i(s_i) ds_i:=\gamma ^*\big ({dt \over t - \zeta ^i}\big ).\end{equation}The ordinary line integral of this 1-form is given by\begin{equation}\int _{\gamma } {dt \over t - \zeta ^i} = \int _a^b f_i(s_i)ds_i\end{equation}which is independent of the parametrization of \)\(. Now choose some r-tuple \) (i1, ..., ir) (5)r\( and consider the iterated integral\begin{equation}L_a(\zeta ^{i_1}, \dots , \zeta ^{i_r}|b): = \int _a^b f_{i_r}(s_{i_r}) \bigg (\int _a^{s_{i_r}} f_{i_{r-1}}(s_{i_{r-1}}) \dots \bigg (\int _a^{s_{i_2}} f_{i_1}(s_{i_1}) ds_{i_1}\bigg )\dots ds_{i_{r-1}}\bigg ) ds_{i_{r}}\end{equation}where \)\(f_{i_r}(s_{i_r}) ds_{i_r}=\gamma ^*\big ({dt \over t - \zeta ^{i_r}}\big ).\)\(This integral is independent of the choice of the parametrization of \)\( and depends on the homotopy class of \)\(. Such integrals, under the name of Hyperlogarithms, were introduced in \cite {Poi} and extensively studied in \cite {LD}.\) The iterated integrals \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| 1)\)  appear in the study of the motivic fundamental group of \(*- \mu _n\)  (when specialized to n=5) carried out in [55]}. These iterated integrals also appear in the construction of the cyclotomic Drinfel'd associator carried out in [56]}.
Let us specialize to the path \([0, 1-\epsilon ] \subset \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\(, where \)\( is an arbitrarily small positive real number. In this case, for any \) (i1, ..., ir) (5)r\(, (\ref {hyperlogii}) is\begin{equation}L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}|1-\epsilon )= \int _0^{1-\epsilon } {1\over s_{i_r} - \zeta ^{i_r}} \bigg (\int _0^{s_{i_r}} {1\over s_{i_{r-1}} - \zeta ^{{i}_{r-1}}} \dots \bigg (\int _0^{s_{i_2}} {1\over s_{i_1} - \zeta ^{{i}_1}} ds_{i_1}\bigg )\dots ds_{i_{r-1}}\bigg ) ds_{i_{r}}.\end{equation}Define \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  [0, 1 - ] End(H0(3, O3\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ))\( by\begin{equation*}\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation*}\) (x) = Id+r=1kr    i1, ..., ir (5)r L0(i1, ..., ir| x)Ai1...Air          x [0, 1 - ],

where we consider \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| x)\)  as a function in one variable: the upper limit of integration.
One easily proves that, see [9]} page 481 or [58]} page 164, that \(\Phi ^{\scriptscriptstyle (}k\) )\(\) (x)\( is convergent for all \) x [0, 1-] \( and\begin{equation}{d\,\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )}\end{equation}_{\epsilon }\over dx\)  = k 1 i 5 \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)   Ai x - i.

Since the connection () is given by the 1-form (), () implies that \(\Phi ^{\scriptscriptstyle (}k\) )\(\)\( gives the parallel transport of the connection (\ref {connectiontildechi}) along the path \) [0, 1 - ]\(.\) The integral () converges as \(\epsilon \rightarrow 0\)  if and only if \(\zeta ^{i_r} \ne 1\) , thus the integral \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| 1)\)  is well defined if and only if \(\zeta ^{i_r} \ne 1\) . Let

\(\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 := Id+r=1kr    i1, ..., ir (5)r ir1 L0(i1, ..., ir| 1)Ai1...Air      End(H0(3, O3\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )).

The endomorphism (REF ) will play a crucial role in the monodromy computation carried out in the next section.
We now give a result on the asymptotic behavior of the parallel transport of () in arbitrarily small neighborhoods of the singularities \(\mu _5\) . Let \(D^*_i:=\lbrace  b_i \in  0 < |b_i| < \epsilon \rbrace \) , and consider the embedding

\(B_i \colon D^*_{i} \rightarrow \tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 
,          bi i + bi .

An easy calculation shows

\(B^*_i (\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
\(\chi \)\(\chi \) ) = k( Ai bi + 1 j 5 i j Aj    bi - (j - i)) dbi.

Let \(Y_i(b_i)\)  be an \(\operatorname{End}(H^0(^3, \mathcal {O}_{^3}(k))) \)  valued function over \(D^*_i\) . Then \(Y_i\)  is a solution of (REF ) if
\(Y_i^{\prime } = \hspace{-5.0pt}k\bigg ( {A_i \over b_i} + \sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} {A_j\,\,\, \over b_i - (\zeta ^j - \zeta ^i)}\bigg ) Y_i.\) 
Proposition 3.6 
For all \(1 \leqslant i \leqslant 5\) , there exists a unique \(Y_i\)  satisfying (REF ) such that
\(Y_i (b_i) = Q_i(b_i) \,b_i^{\hspace{-5.0pt}kA_i}\) 
where \(Q_i(b_i) = \sum _{r\geqslant 0} q^{\scriptscriptstyle (}i\) )\(\) r bir\(, with \) qr\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\)  End(H0(3, O3(k)))\(, and \) Qi(0)=q0\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\)  = Id\(.\)
Remark 3.7 For the expression \(b_i^{\hspace{-5.0pt}kA_i}\)  to make sense on \(D^*_i\) , we must choose a branch cut for the logarithm. We make the choice of the positive real axis in \(D^*_i\)  as the branch cut for the logarithm.

By the theory of ordinary differential equations, see [59]}, there exists a solution \(Y_i\)  satisfying (REF ). We show that there exits a family \(q_r^{\scriptscriptstyle (}i\) )\(\)\( satisfying the above stated requirements such that \) Yi\( is of the form (\ref {localsolution}). We write out\begin{equation}Y_i^{\prime }(b_i)= \bigg (Q_i^{\prime }(b_i) + \hspace{-5.0pt}k{Q_i(b_i)A_i \over b_i}\bigg ) b_i^{\hspace{-5.0pt}kA_i}=\hspace{-5.0pt}k\bigg ({A_i \over b_i} + \sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} {A_j\,\,\, \over b_i - (\zeta ^j - \zeta ^i)}\bigg ) Q_i(b_i) \,b_i^{\hspace{-5.0pt}kA_i}.\end{equation}This can be rewritten as\begin{equation}b_i Q_i^{\prime }(b_i) - \hspace{-5.0pt}k[A_i, Q_i(b_i)]= -\hspace{-5.0pt}k\,\,\sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} A_j {b_i \,Q_i(b_i) \over (\zeta ^j - \zeta ^i)- b_i}.\end{equation}Expanding the above in powers of \) bi\(, we get \) [Ai, q0\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) ]=0\(, and for \) r >0\(\begin{equation}r q^{\scriptscriptstyle (}i\scriptscriptstyle )\end{equation}\) r - k[Ai, q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r] = - kl=1r (1 j 5 i j Aj(j - i)l )q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) l-1

The above equations, for every \(r\) , have a solution. Indeed, if we take \(q^{\scriptscriptstyle (}i\) )\(\) 0=Id\(, then \) q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r\( is uniquely determined by \) q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) 0, ..., q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r-1\( due to the fact that the operator \) r Id- k  ad(Ai)\( is invertible with inverse equal to\begin{equation}{1 \over r} \sum _{n \geqslant 0} {\hspace{-5.0pt}k^n \over r^n} \operatorname{ad}(A_i)^n.\end{equation}The convergence of \) Qi(bi)\( results from the general fact that a formal solution of a regular singular equation is necessarily convergent, see \cite {Kass} and \cite {Was}.\)
Generators of the (orbifold) fundamental group
The curve \(\mathchoice{\chi }{}{}{}\)\(\chi \)\( is defined as the quotient \)\(\chi \)\(\chi \) /D\(, where \) D\( was generated by the automorphisms \) R, I \(\chi \)\(\chi \)  \(\chi \)\(\chi \)\(. The rotation \) R\( is of order five and has two fixed points \) 0\( and \)\(, the involution \) I\( has the fixed point \) -1\(, it follows that \)\(\chi \)\(\chi \)\( is topologically a sphere with two orbifold points \) a := \(\chi \)\(\chi \) (0)\(, \) b:= \(\chi \)\(\chi \) (-1)\( and one puncture. The stabilizer group of the orbifold point \) a\( is \) R\(, and the stabilizer group of the orbifold point \) b\( is \) I\(.\)
We choose explicit paths based at the order five orbifold point of \(\mathchoice{\chi }{}{}{}\)\(\chi \)\( such that they generate the group \) 1orb(\(\chi \)\(\chi \) , a)\(. Let \) 0 \(\chi \)\(\chi \)\( be the path starting from \) 0\( and running along the real axis to \) 1-\(, where \)\( is an arbitrarily small positive real number. Let \) 1\( be the semi-circle starting from \) 1- \( moving around \) 1\( in an anti clockwise direction until it reaches \) (1 - )-1\(. We will also need \) 0-1\( which is the path running from \) 1-\( to \) 0\( along the real axis.\) The end points of the semi-circle \(\tilde{\gamma }_1\)  are identified with each other under the involution \(I\) , which implies that the projection 
\(\pi _{\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }\) 
(1)=:1 \(\chi \)\(\chi \)
\( is a homotopically non-trivial closed loop. Consider the projection of the path \) 
0\(, \)\(\pi _{\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }\) (0)=: 0 \(\chi \)\(\chi \)
\( then the composition\begin{equation}\gamma _0^{-1}\cdot \gamma _1\cdot \gamma _0=: \gamma \subset \mathchoice{\chi }{}{}{}\end{equation}{\chi }\) 

is a closed, connected, homotopically non-trivial loop with the property that it starts and ends at the order five orbifold point \(a \in \mathchoice{\chi }{}{}{}\)\(\chi \)\(.\)
The following is part of proposition 4.11 from [53]}.
Proposition 4.1 
The loop \(\gamma \)  and the orbifold stabilizer group \(\langle R \rangle \)  together generate \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a)\(, and under the isomorphism \) 1orb(\(\chi \)\(\chi \) , a) (2, 5, )\(R is identified with \) U:=ST\( and \)\( is identified with \) T\(.\) 
Let

\(H : \pi _1^{orb}(\mathchoice{\chi }{}{}{}\) 
\(\chi \) , a) H1((a), Z)Sp(4, Z)

be the monodromy representation provided by the Gauss-Manin connection in the bundle of the first de Rahm cohomology groups of the genus two surfaces parametrized by \(\mathchoice{\chi }{}{}{}\)\(\chi \)\(.\begin{lemma}The elements \langle R\rangle  and \gamma  map to M_0 and M_1 respectively where\begin{equation}M_{0}:=\begin{bmatrix}0 & 0 & -1 & 1\\0& 0 & 0 & -1\\1 & 0 & -1 & 0\\1 & 1 & -1 & 0\end{bmatrix} \quad M_{1}:=\begin{bmatrix}0 & 1& 0 & 0\\1&0 & 0 & 0\\1 & 1 & 0 & -1\\1& 1 & -1 & 0\end{bmatrix}.\end{equation}under the homomorphism H.\end{lemma}{\begin{xmlelement*}{proof}Let X_{\sqrt{5}} be the Hilbert modular surface for totally real quadratic field \mathbb {Q}(\sqrt{5}) considered in \cite {CMZ}. The surface X_{\sqrt{5}} is the moduli space of abelian surfaces which admit real multiplication by the field \mathbb {Q}(\sqrt{5}). In \cite {CMZ} the image of Gauss-Mannin connection H^{\prime }: \pi ^{orb}_1(X_{\sqrt{5}}, *) \rightarrow H_1( A_*, \mathbb {Z})\cong \mathrm {Sp}(4, \mathbb {Z}) is described explicitly. In \cite {Mc} it is shown that the Teichmüller curve \mathchoice{\chi }{}{}{}\end{xmlelement*}{\chi }}{\chi }{\chi }\hookrightarrow X_{\sqrt{5}}\)  embeds in the hilbert modular surface; In particular an embedding \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a) 1orb(X5, *)\( is provided. We simply look at the image of \) <R>\( and \)\( under the composition of this embedding with \) H'\(.\)
Let us choose \(\phi (a) \in \mathcal {M}_2\)  as the base point for the orbifold fundamental group of \(\mathcal {M}_2\) . Then \(\phi (\gamma ) \in \pi _1^{orb}(\mathcal {M}_2, \phi (a))\) . In fact, see [40]} or [53]}, the loop \(\phi (\gamma )\)  corresponds to a Dehn twist along two non-separating simple closed loops on a genus two surface, and \(R\)  corresponds to an order five diffeomorphism.
For each integer \(k >0\)  we have the projective vector bundle with the flat connection. The monodromy of this flat connection provides us with a group homomorphism

\(\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 1orb(M2, (a)) End(PV\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (a))

where the fiber \( \operatorname{End}(\mathbb {P}\overline{\mathbb {V}}^{\scriptscriptstyle (}k\) )\(\) (a))\( can be identified with \) End(PH0(3, O3(k)))\(. In the next section, we compute the image of \) ()\( and \) R\( under the morphism (\ref {monodromyrep}), and prove theorem \ref {maintheorem}.\)
Computing Monodromy
In order to compute the image of \(\phi (\gamma ) \)  and \(R\)  under (REF ) we need two connected paths, \(\tilde{\gamma }\)  and \(p_R\) , based at some point \(\bf {z}\in \mathcal {C}\)  such that \(\pi _{\mathcal {M}_2}(\)\(\bf {z}\)\()=\phi (a)\)  and \(\pi _{\mathcal {M}_2}(\tilde{\gamma })=\phi (\gamma )\) , \(\pi _{\mathcal {M}_2}(p_R)=\phi (a)\) .
We have the following well known isomorphism, see [63]} page 188,
\(\pi _1^{orb}(\mathcal {M}_2, *)/(\mathbb {Z}/2) \cong SB_6\) 
where \(\mathbb {Z}/2=\langle \sigma \rangle \)  and \(\sigma \)  is the hyperelliptic involution.
Consider the following map
\(p^{\prime }_I \colon [1, (1-\epsilon )^{-1}] \rightarrow \mathbb {P}\mathrm {SL}(2,  \quad s \mapsto \begin{bmatrix} s^{1\over 2} & 0\\ 0 & s^{-{1\over 2}} \end{bmatrix}.\) 
Let \(p_I \subset \mathcal {C}\)  be the path obtained by applying the family of Möbius transformations \(p^{\prime }_I\)  to the point \(\tilde{\psi }(1-\epsilon ) \in \mathcal {C}\) .
Proposition 4.2 The following
\(\tilde{\gamma }:=\nu ^{\prime }(\tilde{\psi }((\tilde{\gamma }_0)^{-1})))\cdot p_I \cdot \tilde{\psi }(\tilde{\gamma }_1) \cdot \tilde{\psi }(\tilde{\gamma }_0) \subset \mathcal {C}\) 
is a connected path with initial point \(\tilde{\psi }(0)\) , moreover \(\pi _{\mathcal {M}_2}(\tilde{\gamma }) = \phi (\gamma )\) .

Let us first analyze the image \(\tilde{\psi }(\tilde{\gamma }_0)\) . The initial point of the path \(\tilde{\psi }(\tilde{\gamma }_0)\)  is \(\tilde{\psi }(0)\)  which is shown in Figure REF . Here the six points form a symmetric configuration by taking the positions at the fifth roots of unity and zero in the specified order.
<FIGURE>In Figure  the dashed lines show the trajectory of the six points as one travels along \(\tilde{\psi }(\tilde{\gamma }_0)\) . At the end point of \(\tilde{\psi }(\tilde{\gamma }_0)\) , the points \(z_1\)  and \(z_4\)  are a small distance (depends on \(\epsilon \) ) away from the positive real number \({1 \over 2 \mathrm {Re} (\zeta )}\)  and a quick calculation shows that at the end points \(z_1\)  and \(z_4\)  are conjugate to each other, and both have their real parts bigger than \({1 \over 2 \mathrm {Re} (\zeta )}\) . Moreover, \(z_1\)  lies in lower half plane of \( and \) z4\( lies in the upper half plane. This local picture at \) 1 2 Re ()\( is shown in Figure \ref {Fig3}. A similar story holds for the pair \) z2\( and \) z3\( around the negative real number \) 1 2 Re (2)\(. The point \) z5\( moves to \) 12\( and \) z6\( remains fixed at zero. Notice also that the end point of \) (0)\( is the initial point of \) (1)\(.\begin{figure}[ht]\includegraphics [height=3in,width=5in,angle=0]{loops}\caption {}\end{figure}We now look at the configuration at the end point of \) (1)\(. Recall that \) 1\( was a semi-circle traversed anti-clockwise around one from \) (1- )\( to \) (1-)-1\(. At the end point of \) (1)\(, the points \) z1\( and \) z4\( are still conjugate to each other but both have real parts smaller than \) 1 2Re()\( and \) z1\( lies in the upper half plane and \) z4\( lies in the lower half plane. In effect, both \) z1\( and \) z4\( move in an anti-clockwise semi-circle around \) 12Re()\(. This is also shown in figure \ref {Fig3}. A similar story holds for the pair \) z2\( and \) z3\( around the negative real number \) 1 2Re(2)\(. The point \) z5\( moves to \) 12\( and \) z6\( remains fixed at zero. Notice also that the end point of \) (1)\( is not the initial point of \) (0)-1\(. However, initial point of \) pI\( is the end point of \) (1)\( and it follows from\begin{equation*}\tilde{\psi }(1-\epsilon )= \nu ^{\prime } (1-\epsilon )^{-1} (\tilde{\psi }(1-\epsilon )^{-1})\end{equation*}that the configuration at the end point of \) pI\( differs from the configuration at the initial point of \) (0)-1\( only by the permutation \) '\(.\) The trajectory of \(z_4\)  and \(z_1\)  along the path \(p_I\)  is shown in red in figure REF . A similar story holds for \(z_2\)  and \(z_3\) . The point \(z_5\)  moves a little to the right and \(z_6\)  is fixed at zero.
<FIGURE>Since the end point of \(p_I\)  differs from the initial point of \((\tilde{\psi }(\tilde{\gamma }_0)^{-1})\)  by the permutation \(\nu ^{\prime }\) , it follows that \(\nu ^{\prime }(\tilde{\psi }(\tilde{\gamma }_0)^{-1}))\cdot p_I\)  is a connected path with initial point \(\tilde{\psi }((1-\epsilon )^{-1})\)  and end point \(\nu ^{\prime }(\tilde{\psi }(0)\) .
Since the initial point of \(\nu ^{\prime }(\tilde{\psi }(\tilde{\gamma }_0)^{-1}))\cdot p_I\)  is the end point of \(\tilde{\psi }(\tilde{\gamma }_1)\cdot \tilde{\psi }(\tilde{\gamma }_0)\)  it follows that \(\tilde{\gamma }\)  is a connected path in \(\mathcal {C}\)  with initial point \(\tilde{\psi }(0)\)  and end point \(\nu ^{\prime }(\tilde{\psi }(0))\) .
Since \(p_I\)  is entirely contained in the \(\mathbb {P}\mathrm {SL}(2, \)  orbit, it follows that \(\pi _{\mathcal {M}_2} (p_I)\)  is a constant loop in \(\mathcal {M}_2\) . This fact combined with lemma REF  implies
\(\pi _{\mathcal {M}_2}(\tilde{\gamma })=\phi (\gamma ).\) 
Consider the following path in \(\mathbb {P}\mathrm {SL}(2, \)
\(p_R \colon [0, {1 \over 5}] \rightarrow \mathbb {P}\mathrm {SL}(2, , \quad \quad s \mapsto \begin{bmatrix} e^{- \pi i s} & 0 \\ 0 & e^{ \pi i s} \end{bmatrix}\,\,\, s\in [0, {1 \over 5}].\) 
Denote also by \(p_R\)  the path in \(\mathcal {C}\)  given by the action of \(p_R\)  on the point \(\psi (0)\) .
Lemma 4.3 
The path \(p_R \subset \mathcal {C}_{\infty }\)  is connected with initial point \(\widetilde{\psi }(0)\) , moreover \(\pi _{\mathcal {M}_2}(p_R)= \phi (a)\) .

At both the initial and the end point of this path the first five points sit at the fifth roots of unity (sixth at zero) but with different ordering. The ordering differs by \(\nu ^{-1}\) . Since the end points of \(p_R\)  are related by an element in \(S_6\) , \(P_{\infty }(p_R) \subset \overline{\mathcal {C}}\)  is a closed loop. Moreover, since \(p_R\)  is entirely contained in the \(\mathbb {P}\mathrm {SL}(2, \)  orbit, \((\overline{P}_{\infty }\cdot P_{\infty })(p_R)\)  is a constant path in \(\mathcal {M}_2\) , namely \(\phi (a)\) .
<FIGURE>We now prove theorem REF  from the introduction.
(Theorem REF )
We compute the parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along \)\( and \) pR\(. Recall from (\ref {tildegamma})\begin{equation}\tilde{\gamma }:=\nu ^{\prime }(\psi ((\tilde{\gamma }_0)^{-1})))\cdot p_I \cdot \psi (\tilde{\gamma }_1) \cdot \psi (\tilde{\gamma }_0) \subset \mathcal {C}\end{equation}and \) pR\( is given in lemma \ref {60}.\)
We first observe the following
Lemma 4.4 The parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along the paths \) pI\( and \) pR\( is projectively trivial.\)
Recall that \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C:= d + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\( and \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\( is projectively invariant under the \) PSL(2, \( on \) C\(. Lemma follows since both \) pR\( and \) pI\( are contained in the orbit of the dilation subgroup of \) PSL(2, \(.\)
In order to compute the parallel transport along \(\tilde{\gamma }\)  we must compute the parallel transport along all the paths separately appearing in the expression above. We compute the parallel transport along \(\widetilde{\psi }(\tilde{\gamma }_1)\) . The semi circle \(\tilde{\gamma _1} \subset \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( is contained in the image of \) B1\(, defined in (\ref {discs}), which is a punctured disk of radius \)\( based at \) 1 \(\chi \)\(\chi \)\(.\)
Proposition 4.5 
The parallel transport along \((\widetilde{\psi }\circ B_1)^{*}(\psi (\tilde{\gamma }_1))\)  with respect to \((\widetilde{\psi }\circ B_1) ^{*}\nabla ^{\scriptscriptstyle (}k\) )\(\)\( is\begin{equation}Q_1(-i\epsilon )\cdot \exp \big (-\pi i \hspace{-5.0pt}kA_1 \big )\cdot Q_1(i \epsilon ),\end{equation}where \) Q1(b1)\( is the power series from proposition \ref {localsolutionproposition}.\)
Remark 4.6 We assume the \(\epsilon \)  in the definition of \(B_1\)  to be bigger than the \(\epsilon \)  in the definition of \(\tilde{\gamma }_1\) . The \(\epsilon \)  appearing in the proposition refers to the \(\epsilon \)  for \(\tilde{\gamma }_1\) .

From proposition REF  we know that the parallel transport of
\((\widetilde{\psi }\circ B_1 )^{*}\nabla ^{\scriptscriptstyle (}k\) )\(\)\( is given by a solution of the form \) Y1(b1)\( shown in (\ref {localsolution}). The pull back \) (B1)*((1))\( is a semi circle in \) D1\( traveling in an anti-clockwise direction from \) i\( to \) -i \(, avoiding the positive real axis in \) D1\( (which was our choice for the branch cut of the logarithm on \) D1\(). Thus the parallel transport is given by{\begin{@align*}{1}{-1}Y_1(-i\epsilon )(Y_1(i\epsilon ))^{-1} &= Q_1(-i\epsilon ) (- i \epsilon )^{\hspace{-5.0pt}kA_1} (i\epsilon ^{-\hspace{-5.0pt}kA_1})Q_1(i\epsilon )^{-1} \\& = Q_1(-i\epsilon ) \exp (log(-i\epsilon ) \hspace{-5.0pt}kA_1)\exp (-log (i\epsilon ) \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1}\\&= Q_1(-i\epsilon ) \exp ((log(i\epsilon ) + \pi i)\hspace{-5.0pt}kA_1) \exp (-log (i\epsilon ) \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1}\\&=Q_1(-i\epsilon ) \exp (\pi i \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1},\end{@align*}}where the first equality is from (\ref {localsolution}) and third equality follows from our choice of the branch cut for the logarithm.\)
For the parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along \) (0)\(, where recall that \) 0 \(\chi \)\(\chi \)\( is the interval \) [0, 1 - ]\(, we take the pull back \) *(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )\(, which is given in (\ref {psipullback}), restrict it to the interval \) [0, 1 - ]\(, and compute the monodromy there. This monodromy was computed in section \ref {iteratedintegralsection} and is given by \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1- )\( c.f. (\ref {30}).\)
Now, it follows that under the map (REF ) we have that

\(\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
(R) = M\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0

since parallel transport along \(p_R\)  is projectively trivial. Here \(M_0^{\scriptscriptstyle (}k\) )\(\)\( denotes the \) k\(-th symmetric power of the endomorphism \) M0\( from lemma \ref {m0m1}.Likewise we have that{\begin{@align*}{1}{-1}\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{@align*}}(\phi (\gamma )) &=& M^{\scriptscriptstyle (}k\) )\(\) 1 ' (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1 Q1(-i) exp(-i kA1 )(Q1(i ))-1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - )
= (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1 M1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (Q1(-i) exp(-i kA1 )Q1(i )-1)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - )

where we used the fact that \(M_1^{\scriptscriptstyle (}k\) )\(\)\( commutes with the parallel transport, see \cite {GJ}.This commutation also had the effect ofremoving \) '\( from\) '(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1)\( because \) M1\( maps to \) '\(under the map \) A(G) S6\(.\)
In [56]}, see also [9]} and [58]}, it is shown that
\(&\lim _{\epsilon \rightarrow 0}&\Big ((\Phi _{\epsilon }^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
(1 - ))-1 M1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (Q1(-i) exp(-i kA1 )Q1(i )-1)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))
= (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )-1((M\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 1)-1exp( -i kAi)) \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)
= \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (())

where \(\Phi ^{\scriptscriptstyle (}k\) )\(\)\( is defined in (\ref {911}).\) Replacing \(R\)  with \(U\)  and \(\phi (\gamma )\)  with \(T\)  under the isomorphism \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a) (2, 5, )\( we get the statement of theorem \ref {maintheorem}.\)
